3.652 \(\int \cos (x) \cos (\cos (x)) \sin (x) \sin (\cos (x)) \, dx\)

Optimal. Leaf size=28 \[ \frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x)) \]

[Out]

Cos[x]/4 - (Cos[Cos[x]]*Sin[Cos[x]])/4 - (Cos[x]*Sin[Cos[x]]^2)/2

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Rubi [A]  time = 0.0255663, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4335, 3443, 2635, 8} \[ \frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Cos[Cos[x]]*Sin[x]*Sin[Cos[x]],x]

[Out]

Cos[x]/4 - (Cos[Cos[x]]*Sin[Cos[x]])/4 - (Cos[x]*Sin[Cos[x]]^2)/2

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos (x) \cos (\cos (x)) \sin (x) \sin (\cos (x)) \, dx &=-\operatorname{Subst}(\int x \cos (x) \sin (x) \, dx,x,\cos (x))\\ &=-\frac{1}{2} \cos (x) \sin ^2(\cos (x))+\frac{1}{2} \operatorname{Subst}\left (\int \sin ^2(x) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x))-\frac{1}{2} \cos (x) \sin ^2(\cos (x))+\frac{1}{4} \operatorname{Subst}(\int 1 \, dx,x,\cos (x))\\ &=\frac{\cos (x)}{4}-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x))-\frac{1}{2} \cos (x) \sin ^2(\cos (x))\\ \end{align*}

Mathematica [A]  time = 1.51206, size = 21, normalized size = 0.75 \[ \frac{1}{4} \cos (x) \cos (2 \cos (x))-\frac{1}{8} \sin (2 \cos (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cos[Cos[x]]*Sin[x]*Sin[Cos[x]],x]

[Out]

(Cos[x]*Cos[2*Cos[x]])/4 - Sin[2*Cos[x]]/8

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Maple [A]  time = 0.009, size = 23, normalized size = 0.8 \begin{align*}{\frac{ \left ( \cos \left ( \cos \left ( x \right ) \right ) \right ) ^{2}\cos \left ( x \right ) }{2}}-{\frac{\cos \left ( \cos \left ( x \right ) \right ) \sin \left ( \cos \left ( x \right ) \right ) }{4}}-{\frac{\cos \left ( x \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cos(cos(x))*sin(x)*sin(cos(x)),x)

[Out]

1/2*cos(cos(x))^2*cos(x)-1/4*cos(cos(x))*sin(cos(x))-1/4*cos(x)

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Maxima [A]  time = 0.96901, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{4} \, \cos \left (x\right ) \cos \left (2 \, \cos \left (x\right )\right ) - \frac{1}{8} \, \sin \left (2 \, \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(cos(x))*sin(x)*sin(cos(x)),x, algorithm="maxima")

[Out]

1/4*cos(x)*cos(2*cos(x)) - 1/8*sin(2*cos(x))

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Fricas [B]  time = 2.05015, size = 219, normalized size = 7.82 \begin{align*} \frac{1}{2} \, \cos \left (x\right ) \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{2} + \frac{1}{4} \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) - \frac{1}{4} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(cos(x))*sin(x)*sin(cos(x)),x, algorithm="fricas")

[Out]

1/2*cos(x)*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))^2 + 1/4*cos((tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1))*sin((
tan(1/2*x)^2 - 1)/(tan(1/2*x)^2 + 1)) - 1/4*cos(x)

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Sympy [A]  time = 7.89546, size = 34, normalized size = 1.21 \begin{align*} - \frac{\sin ^{2}{\left (\cos{\left (x \right )} \right )} \cos{\left (x \right )}}{4} - \frac{\sin{\left (\cos{\left (x \right )} \right )} \cos{\left (\cos{\left (x \right )} \right )}}{4} + \frac{\cos{\left (x \right )} \cos ^{2}{\left (\cos{\left (x \right )} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(cos(x))*sin(x)*sin(cos(x)),x)

[Out]

-sin(cos(x))**2*cos(x)/4 - sin(cos(x))*cos(cos(x))/4 + cos(x)*cos(cos(x))**2/4

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Giac [A]  time = 1.08866, size = 23, normalized size = 0.82 \begin{align*} \frac{1}{4} \, \cos \left (x\right ) \cos \left (2 \, \cos \left (x\right )\right ) - \frac{1}{8} \, \sin \left (2 \, \cos \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cos(cos(x))*sin(x)*sin(cos(x)),x, algorithm="giac")

[Out]

1/4*cos(x)*cos(2*cos(x)) - 1/8*sin(2*cos(x))